#8345970

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4074 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,4)(3,8,6,10)(5,9,14,12)(11,15,13,16)(17,18,24,30)(19,31,32,25)(20,27,21,26)(22,23,29,28)(33,34,39,42)(35,41,37,40)(36,45,46,38)(43,44,47,48);(1,3,7,6)(2,8,4,10)(5,11,14,13)(9,15,12,16)(17,19)(18,25)(20,29)(21,22)(23,27)(24,32)(26,28)(30,31)(33,35,39,37)(34,40,42,41)(36,43,46,47)(38,44,45,48);(3,6)(8,10)(11,13)(15,16)(17,20,24,21)(18,26,30,27)(19,29,32,22)(23,25,28,31)(33,35,39,37)(34,41,42,40)(36,43,46,47)(38,48,45,44);(1,4,7,2)(3,10,6,8)(5,9,14,12)(11,15,13,16)(17,21,24,20)(18,27,30,26)(19,29,32,22)(23,25,28,31)(33,36)(34,38)(35,43)(37,47)(39,46)(40,44)(41,48)(42,45);(1,5,7,14)(2,9,4,12)(3,11,6,13)(8,15,10,16)(17,22,24,29)(18,28,30,23)(19,21,32,20)(25,26,31,27)(33,37,39,35)(34,40,42,41)(36,43,46,47)(38,48,45,44);(1,6,7,3)(2,10,4,8)(5,13,14,11)(9,16,12,15)(17,23,24,28)(18,29,30,22)(19,27,32,26)(20,31,21,25)(33,38)(34,36)(35,44)(37,48)(39,45)(40,43)(41,47)(42,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=hi,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8253166);